Mathematics of Cryptography

Winter 2011 - Math 4161 3.0


Tuesday/Thursday  2:30-4pm  CLH M

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Professor Mike Zabrocki
Office: TEL 2028
Office Hours: Monday/Tuesday 4-6pm
e-mail : my e-mail
web page: http://garsia.math.yorku.ca/~zabrocki
course web page: http://garsia.math.yorku.ca/~zabrocki/math4161w11/
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Course Description :Cryptography deals with the study of making and breaking secret codes.

In this course we will be studying situations that are often framed as a game between three parties: a sender (e.g., an embassy), a receiver (the government office) and an opponent (a spy). We assume that the sender needs to get an urgent message to the receiver through communication channels which are vulnerable to the opponent. To do this communication, the sender and receiver agree in advance to use some sort of code which is unlocked by a keyword or phrase. The opponent will be able to intercept the message. Is he/she able to unlock the message without knowing the key?

In this course we will learn some probability theory, information theory and number theory to answer questions about how vulnerable the methods of sending secrets are. This has a great number of applications to internet credit card transactions, wireless communication and electronic voting. We will start by learning some classical codes (used up through WWI) and analyzing those. The last third of the course we will start to learn the methods that are used in modern cryptography.


Announcements:

(January 4, 2011) (RULE #1) I will NEVER EVER post solutions to practice questions or quizzes. Don't ask. Email questions like "what is the answer to number xxxx?" will mostly be ignored. I will answer questions about the practice by email on an individual basis but I need to know what you tried to solve the problem.

(January 20, 2011) So it looks like my calculations for the encrypting matrix was correct (and these have now been posted), but some of my reasoning was a bit wrong. I said, that IF \[ \begin{bmatrix} -5&3\\5&19\end{bmatrix} \begin{bmatrix}a\\b\end{bmatrix} \equiv \begin{bmatrix}17\\9\end{bmatrix}~~(mod~26)\] THEN (by multiplying by the matrix \( \begin{bmatrix} 19&-3\\-5&-5 \end{bmatrix} \) on both sides of the equation) \[ \begin{bmatrix} 20&0\\0&20\end{bmatrix} \begin{bmatrix}a\\b\end{bmatrix} \equiv \begin{bmatrix}10\\0\end{bmatrix}~~(mod~26).\] However, you cannot conclude the reverse implication and you must check which of the four solutions \( (a,b) \in \{ (7,0),(7,13),(20,0),(20,13) \} \) does solve the first set of equations. Only for \( (a,b) \in \{ (7,0), (20, 13) \} \) does it work.

(February 8, 2011) By request I added a few more questions for practice. Check in the handouts section. You should have all gotten a cryptogram to break using the applets. I sent it to the email address that the university has on file for you. If you don't find it, check in the spam box. Read the instructions carefully.

(February 9, 2011) A tentative time has been announced for the final exam of April 19, 2011 from 9am-12pm. In a week or two we will receive confirmation of this.

(February 24, 2011) The University of Toronto is hosting a day of math related to number theory and cryptography to be held Saturday March 5, 2011 from roughly 10am - 4pm. It seems like the schedule will be about cryptography from 10-12am and the afternoon is about topics in number theory. Typically these events are underrepresented by York University students so it would be good if we can drum up some business from our class. Facebook page.

(April 4, 2011) Once classes are over starting April 5 I will not be in my office on the same days as I usually am. Here is a list of my updated office hours:
April 4, 4-6pm
April 14, 2-4pm
April 15, 3-5pm (tentative)
It is a good idea to let me know if you are coming because I may leave if there isn't anyone there the first hour or so.

Text : I will not be following a textbook for the course. The last time I had a textbook I followed class notes much closer than we followed the book. The text that I used last time was 'Cryptography: an introduction' by Nigel Smart.  If you feel like you would like to have a reference book in addition to the class notes that I will provide you with, then I suggest that you search this book out. I expect the notes will prove to be more useful and it is important that you come to class to ask questions. FYI, the book does not cover the introductory material on classical ciphers very well, but I like it.

Issues of Academic Integrity : Your exams and quizzes will be open books and notes. I want you to have access to reference material when you are working. I expect you however to keep your eyes on your own paper. Students are expected to be familar with the Senate Policy on Academic Honesty and to follow it. The one time that I taught this class I had at least two people pass through hearings with the administration because of issues with academic dishonesty. This time I will not take chances and I intend to put as many mesures into effect to stop cheating as possible.

Calculators that use + - * / ^ and log are allowed on quizzes and tests. Calculators which have more advanced functions like factor, gcd, Jacobi, mod and discrete log are not currently commonplace and until those functions are available to everyone I expect you to stick with a basic calculator. No smart phones. Books and notes are allowed on the tests and quizzes as well but I try to add some creative way of making the problems unique so that they are not the same as changing a few numbers from a practice problem. It is not my job to watch over your shoulder to tell you the difference between right and wrong. I give you a lot of leeway (e.g. open book and notes, and calculators) in this class because I expect you to be honest and follow these rules and not copy off of your neighbor when we have tests and quizzes.

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